Internal graphs of graph products of hyperfinite II$_1$-factors

TL;DR

This paper introduces the internal graph as an invariant for classifying graph products of hyperfinite II$_1$-factors, enabling new classification results for specific graph types and their isomorphisms.

Contribution

It defines the internal graph as an isomorphism invariant for hyperfinite II$_1$-factor graph products and applies it to classify certain graph products and analyze their isomorphisms.

Findings

Internal graph is an isomorphism invariant for hyperfinite II$_1$-factor graph products.

Classification of $R_{ ext{graph}}$ for lines, cycles, and infinite regular trees.

Difference in graph radius is at most 1 for isomorphic graph products.

Abstract

In this paper, we show that for a graph $\Gamma$ from a class named H-rigid graphs, its subgraph ${\rm Int}(\Gamma)$, named the internal graph of $\Gamma$, is an isomorphism invariant of the graph product of hyperfinite II$_1$-factors $R_{\Gamma}$. In particular, we can classify $R_{\Gamma}$ for some typical types of graphs, such as lines, cyclic graphs and infinite regular trees. As an application, we also show that for two isomorphic graph products of hyperfinite II$_1$-factors over H-rigid graphs, the difference of the radius between the two graphs will not be larger than 1. Our proof is based on the recent resolution of the Peterson-Thom conjecture.